Method For Elaborating Navigation Parameters And Vertical Of A Place

ABSTRACT

The invention concerns the gyroscopic instrument engineering and can be used in determination of main navigation parameters for marine, air and surface vehicle positioning. Technical result consists in increased accuracy of output parameters, including substantial increase of the vehicle heading generation accuracy as well as principal improvement of dynamic properties of output parameter generation. To achieve these results control signals of gyro platforms or gyro platform models are generated under the condition that in the absence of ballistic deviations, autonomously, a natural period of the gyro platforms or the gyro platform models differ from Schuler period.

The invention concerns the gyroscopic instrument engineering and can be used in navigation of marine, airborne and surface vehicles.

Well-known is the method of navigation parameters and position vertical generation. This method includes measurement of specific force components using accelerometers, sensitivity axes of which are connected with a gyro platform; generation of gyro platform control signals; following to indicate the generated signals by means of a gyroscope or absolute angular rate sensors [1]. Otherwise this method may include measurement of specific force components using accelerometers; measurement of signals from gyros or absolute angular rate sensors, sensitivity axes of which are directed along axes of instrumental trihedral; analytical solution of the orientation problem by means of inertial system operation simulation [2]; generation of navigation parameters and position vertical.

The disadvantage of this method consists in limited capabilities of accuracy and dynamic characteristics.

The purpose of the invention is increase of accuracy characteristics and enhancement of dynamic capabilities of the method.

The technical effect is achieved in such a way:

control signals of a gyro platform or a model of the gyro platform of the inertial system with linear correction are generated under the condition that in the absence of ballistic deviations a natural period of the gyro platform or the gyro platform model differs from Schuler period;

control signals of gyro platforms or models of gyro platforms of the inertial system with linear correction are generated to provide, in the absence of ballistic deviations, either nonlinear relation between a value of speed deviation and a value of horizontal component of the vehicle absolute angular rate, or the relation of $\alpha = {\frac{n}{\omega_{0}}\frac{V}{R}}$ type at small values of speed deviations, with which relations tgα=sin α=α are admissible;

asymptotic stability (autonomous damping) of each inertial system or each inertial system model, including (if necessary) the inertial system with integral correction, is provided using differences of the similar data generated by inertial systems with linear correction, as well as the inertial system with integral correction (if necessary), or their models with different natural periods and (or) using values of observable generalized coordinates of the system or its model;

estimation of instrumental errors is provided using differences of the similar data generated by inertial systems with linear corrections, as well as the inertial system with integral correction (if necessary), or their models with different natural periods (with different parameters “n”) and in different modes;

accuracy of the vehicle heading generation is increased and readiness time is reduced due to increasing directing force to the compass meridian in the inertial system with linear correction.

The maximum effect in estimation of instrumental errors and improvement of dynamic behavior of the system is provided by analytical solution of the orientation problem in a strapdown system. In this case the similar data from different models of simultaneously operating inertial systems with different parameters and in different modes are compared.

We can present several examples of inertial systems for implementation of the method: an inertial system with linear correction with a gyro platform in two-axis and three-axis gimbals; an inertial system with linear correction in joint operation with an inertial system with integral correction; a strapdown inertial system.

The drawing presents a functional block diagram of the inertial system for implementation of the method (see FIG. 1).

FIGS. 2, 3, 4 show the schemes of gyro platforms with different gimbal types. FIGS. 2 and 3 show the schemes of the gyro platforms in three-axis gimbal. FIG. 4 shows the scheme of the gyro platform in two-axis gimbal.

The inertial system with linear correction, which is under consideration (FIG. 1, FIG. 2, FIG. 3), consists of two structurally identical stabilized gyro platforms 1 and 1′ and unit 2—a unit of control and output parameter generation. One two-degree-of-freedom gyroscope (3 and 3′) is located on each stabilized gyro platform. And with this, angular momentums of the gyros are perpendicular to the planes of the stabilized gyro platforms. Gyroscopes have torquers 4, 5 and 4′, 5′ and angular pick-offs 6, 7 and 6′, 7′. Besides, accelerometers 8, 9 and 8′, 9′ are mounted on each stabilized gyro platform. Accelerometer sensitivity axes on each gyro platform are orthogonal between each other and are parallel to the gyro platform plane. The axis of one accelerometer is parallel to the inner axis of the gyro platform gimbal. The outer axes of gimbals 11 and 11′ are installed in the common gimbal ring 18 (see FIG. 2). The axis of the common gimbal ring is installed on the horizon-stabilized platform 19 of the inertial system with integral correction. Axes 11 and 11′ are parallel to the plane of the stabilized platform 19 and are parallel to each other. Motor 20 and heading sensor 21 are located on the common gimbal ring axis. Angular pick-offs 16 and 16′, measuring speed deviations α₁ and α₂, are also located on the common ring 18.

By means of amplifiers 12, 13 and 12′, 13′ outputs of angular pick-offs 6, 7 and 6′, 7′ of gyroscopes 3 and 3′ are connected with inputs of motors 14, 15 and 14′, 15′, which are connected with the gimbal axes. Angular pick-offs 16, 17 and 16′, 17′ are connected with the same axes. Inputs of torquers 4, 5 and 4′, 5′ of gyros 3 and 3′ are connected with the corresponding outputs of control and output parameter generation unit 2. Outputs of accelerometers 8, 9 and 8′, 9′ and angular pick-offs 16, 17 and 16′, 17′ are connected with the corresponding inputs of unit 2. Unit 2 has data communication with the inertial system with integral correction. Outputs of unit 2 for users are K—vehicle heading, φ—latitude, λ—longitude, θ and ψ—roll and pitch angles.

The proposed system operates as follows. With the help of motors 14, 15 and 14′, 15′ respectively by error signals from angular pick-offs 6, 7 and 6′, 7′ of gyros 3 and 3′ each gyro platform is constantly kept in one plane with the gyro housing.

The housing of each gyro together with the gyro platform is brought to the position corresponding to the prescribed value of speed deviation for the given gyro platform with the help of torques applied via torquers 4, 5 and 4′, 5′ of gyros 3 and 3′ by control currents by signals generated in unit 2. Since the prescribed values of speed deviations are different for each gyro platform, differences in readings of angular pick-offs 16 and 16′ are the initial data sources for determination of horizontal component of absolute angular rate of Darboux trihedral. Using servo motor 20 the plane of the common gimbal ring 18 is constantly kept in direction perpendicular to the plane of compass meridian.

Let's choose Darboux E₀N₀ζ₀ moving trihedral, axis ON₀ of which is oriented by the horizontal component of absolute angular rate V/R,

as the initial coordinate system.

Then projections of absolute angular rate of the trihedral E₂N₂ζ₂ on its axes will be 0; V/R;

r. Projections of trihedral vertex acceleration E₀N₀ζ₀ on its axes OE₀ and ON₀ are {dot over (V)}; rV.

Projection of gravity acceleration on axis Oζ₀ will be ${- g} = {- {\left( {g_{0} - \frac{V^{2}}{R}} \right).}}$

For the schemes in FIGS. 2 and 3 let the right coordinate system E₁N₁ζ₁ be rigidly bound with the gyro housing of the first gyro platform. Let coordinate system E₂N₂ζ₂ be rigidly bound with the gyro housing of the second gyro platform. Coordinate system E₁N₁ζ₁ will be obtained by turning around axis OE₀ at angle α₁ and around auxiliary axis ON₁′ at angle β₁. Coordinate system E₂N₂ζ₂ will be obtained by turning around axis OE₀ at angle α₂ and around auxiliary axis ON₂′ at angle β₂.

Projections of absolute angular rate of trihedrals E₁N₁ζ₁ and E₂N₂ζ₂ on their axes OE₁; ON₁; OE₂; ON₂, taking account of gyro drifts, will be $\Omega_{E\quad 1} = {{\overset{.}{\alpha}}_{1} + {\left( {{\frac{V}{R}\sin\quad\alpha_{1}} - {r\quad\cos\quad\alpha_{1}}} \right)\beta_{1}} + {\Delta\quad p_{1}}}$ $\Omega_{N\quad 1} = {{\overset{.}{\beta}}_{1} + {\frac{V}{R}\cos\quad\alpha_{1}} + {r\quad\sin\quad\alpha_{1}} + {\Delta\quad q_{1}}}$ $\Omega_{E\quad 2} = {{\overset{.}{\alpha}}_{2} + {\left( {{\frac{V}{R}\sin\quad\alpha_{2}} - {r\quad\cos\quad\alpha_{2}}} \right)\beta_{2}} + {\Delta\quad p_{2}}}$ $\Omega_{N\quad 2} = {{\overset{.}{\beta}}_{2} + {\frac{V}{R}\cos\quad\alpha_{2}} + {r\quad\sin\quad\alpha_{2}} + {\Delta\quad q_{2}}}$ where Δp₁; Δp₂; Δq₁; Δq₂—gyro drifts.

Accelerometer signals by axes OE₁; ON₁ and axes OE₂; ON₂ will be the following: W _(E1) =−V−(rV sin α₁ −g cos α₁)β₁ +ΔW _(E1) W _(N1)=−(rV cos α₁ +g sin α₁)+ΔW _(N1) W _(E2) =−V−(rV sin α₂ −g cos α₂)β₂ +ΔW _(E2) W _(N2)=−(rV cos α₂ +g sin α₂)+ΔW_(N2) where ΔW_(E1); ΔW_(N1); αW_(E2); αW_(N2)—accelerometer errors.

In order to provide the gyro platform natural period that differs from Schuler period, invariant values of speed deviations can be as follows, for example: $\begin{matrix} {{{{tg}\quad\alpha_{1}} = {\frac{n_{1}}{\omega_{0}}\frac{V}{R}}};} & {{{tg}\quad\alpha_{2}} = {\frac{n_{2}}{\omega_{0}}\frac{V}{R}}} & {or} \\ {{{\sin\quad\alpha_{1}} = {\frac{n_{1}}{\omega_{0}}\frac{V}{R}}};} & {{\sin\quad\alpha_{2}} = {\frac{n_{2}}{\omega_{0}}\frac{V}{R}}} & {or} \end{matrix}$ have the form of other nonlinear relations between a value of speed deviation α and a value of horizontal component of absolute angular rate V/R, where

-   -   ω₀—Schuler frequency;     -   n₁; n₂—system parameters.

As an example, let us consider the behavior of the inertial system with linear correction for the case ${{{tg}\quad\alpha_{1}} = {\frac{n_{1}}{\omega_{0}}\frac{V}{R}}};{{{tg}\quad\alpha_{2}} = {\frac{n_{2}}{\omega_{0}}{\frac{V}{R}.}}}$

Let's assume that n₂=−n₁=n.

In order to provide invariant values of speed deviations for this case, gyro control signals can have the form, for example, ${\Omega_{E\quad 1\quad{{con}.}} = {\frac{n}{\omega_{0}}\left( {\frac{W_{E\quad 1}}{R} + {\frac{{\overset{.}{V}}_{{dev}.}}{R}\sin^{2}\alpha_{{dev}.}}} \right)}};$ ${\Omega_{E\quad 2\quad{{con}.}} = {{- \frac{n}{\omega_{0}}}\left( {\frac{W_{E\quad 2}}{R} + {\frac{{\overset{.}{V}}_{{dev}.}}{R}\sin^{2}\alpha_{{dev}.}}} \right)}};$ ${\Omega_{N\quad 1\quad{{con}.}} = {{\frac{\omega_{0}}{n}\sin\quad\alpha_{{dev}.}} + {\frac{n}{\omega_{0}}\frac{\left( {W_{N\quad 1} + W_{N\quad 2}} \right)}{2\quad R}}}};$ $\Omega_{N\quad 2\quad{{con}.}} = {{\frac{\omega_{0}}{n}\sin\quad\alpha_{{dev}.}} - {\frac{n}{\omega_{0}}\frac{\left( {W_{N\quad 1} + W_{N\quad 2}} \right)}{2\quad R}}}$ where

2α_(dev.)=α₂−α₁—difference in readings of angular pick-offs 16′ and 16; ${{\frac{\omega_{0}}{n}\sin\quad\alpha_{np}} = {\frac{V_{np}}{R}\cos\quad\alpha_{np}}};$ ${\frac{n}{\omega_{0}}\frac{{\overset{.}{V}}_{np}}{R}} = {\frac{{\overset{.}{\alpha}}_{np}}{\cos^{2}\alpha_{np}}.}$

The system operation equations will be the following: Ω_(E1)=Ω_(E1con.) ΩE₂=Ω_(E2con.) ΩN₁=Ω_(N1con.) ΩN₂=Ω_(N2con.)

The equations of gyro platform errors or gyro platform model errors, with allowance for the influence of members with vertical component of absolute angular rate r and gyro platform turn around the angular momentum axis at angle ΔK cos α as a result of motor 20 action, will be the following: $\begin{matrix} {{\frac{{- \Delta}\quad\overset{\overset{\sim}{.}}{\alpha}}{\cos^{2}\alpha} + {\frac{r}{\cos\quad\alpha}\overset{\sim}{\beta}} - {n\quad\omega_{0}\cos\quad\alpha\quad\overset{\sim}{\beta}}} = {{{\Delta\quad\overset{\sim}{p}} + {\frac{n}{\omega_{0}R}\Delta\quad{\hat{W}}_{E}} - \overset{\hat{.}}{\beta} - {r\quad\cos\quad\alpha\quad\Delta\quad\hat{\alpha}} + {\frac{\omega_{0}}{n\quad\cos\quad\alpha}\Delta\quad\overset{\sim}{\alpha}}} = {{{\Delta\quad\hat{q}} - \overset{\overset{\sim}{.}}{\beta} - {\frac{r}{\cos\quad\alpha}\Delta\quad\overset{\sim}{\alpha}} + {n\quad\omega_{0}\cos\quad\alpha\quad\Delta\quad\overset{\sim}{\alpha}}} = {{{\Delta\quad\overset{\sim}{q}} + {\frac{n}{\omega_{0}R}\Delta\quad{\hat{W}}_{N}} - {\Delta\quad\overset{\hat{.}}{\alpha}} + {\frac{r}{\cos\quad\alpha}\hat{\beta}} - {n\quad\omega_{0}\cos\quad\alpha\quad\overset{\sim}{\beta}}} = {{\Delta\quad\hat{p}} + {\frac{n}{\omega_{0}R}\Delta\quad{\overset{\sim}{W}}_{E}} + {\left( {1 - n^{2}} \right)\cos^{2}\alpha\quad\frac{V}{R}\Delta\quad K}}}}}} & (1) \end{matrix}$ where ${\frac{{\Delta\quad\alpha_{2}} - {\Delta\quad\alpha_{1}}}{2} = {\Delta\quad\overset{\sim}{\alpha}}};{\frac{{\Delta\quad\alpha_{2}} + {\Delta\quad\alpha_{1}}}{2} = {\Delta\quad\hat{\alpha}}};{\frac{\beta_{2} + \beta_{1}}{2} = \hat{\beta}};{\frac{\beta_{2} - \beta_{1}}{2} = \overset{\sim}{\beta}};$ ${\frac{{\Delta\quad p_{2}} - {\Delta\quad p_{1}}}{2} = {\Delta\quad\overset{\sim}{p}}};{\frac{{\Delta\quad p_{2}} + {\Delta\quad p_{1}}}{2} = {\Delta\quad\hat{p}}};{\frac{{\Delta\quad q_{2}} - {\Delta\quad q_{1}}}{2} = {\Delta\quad\overset{\sim}{q}}};{\frac{{\Delta\quad q_{2}} + {\Delta\quad q_{1}}}{2} = {\Delta\quad\hat{q}}};$ ${\frac{{\Delta\quad W_{E\quad 2}} - {\Delta\quad W_{E\quad 1}}}{2} = {\Delta\quad{\overset{\sim}{W}}_{E}}};{\frac{{\Delta\quad W_{E\quad 2}} + {\Delta\quad W_{E\quad 1}}}{2} = {\Delta\quad{\hat{W}}_{E}}};{\frac{{\Delta\quad W_{N\quad 2}} + {\Delta\quad W_{N\quad 1}}}{2} = {\Delta\quad{\hat{W}}_{N}}}$ $\frac{\beta_{2} - \beta_{1}}{2} = {\overset{\sim}{\beta}\text{-}}$ observable generalized coordinate.

In the strapdown variant measurement (β₂−β₁), as well as measurement (α₂−α₁), are obtained by estimating deflection of the instrumental vertical of each gyro platform of the inertial system with linear correction from the direction of the geocentric vertical of the inertial system with integral correction in the form of angles β and α.

Assuming that ${{{\Delta\quad\Omega_{N\quad 0}} = \frac{\omega_{0}\Delta\quad\overset{\sim}{\alpha}}{n\quad\cos^{2}\alpha_{2}}};{{\Delta\quad\Omega_{E\quad 0}} = {\Omega_{N\quad 0}\frac{\overset{\sim}{\beta}}{\sin\quad\alpha_{2}}}};{{\overset{\sim}{\beta}}_{1} = \frac{\hat{\beta}}{\cos\quad\alpha_{2}}}},$ where

-   -   ΔΩ_(N0); ΔΩ_(E0)—errors in generation of horizontal components         of absolute angular rate of the vehicle;     -   {circumflex over (β)}₁—stabilization error of the horizon plane         around axis ON₀,         we'll obtain equation set (1) in more convenient form:         $\begin{matrix}         {{{{{{{{{{{{{- \Delta}\quad{\overset{.}{\Omega}}_{N\quad 0}} + {r\quad\Delta\quad\Omega_{E\quad 0}} - {\left( {\omega_{0}^{2}\cos^{2}\alpha_{2}} \right){\hat{\beta}}_{1}}} = {\frac{\Delta\quad\overset{\sim}{p}\omega_{0}}{n} + \frac{\Delta\quad{\hat{W}}_{E}}{R}}};} - {\overset{\hat{.}}{\beta}}_{1} - {r\quad\Delta\quad\hat{\alpha}} + {\Delta\quad\Omega_{N\quad 0}}} = \frac{\Delta\quad\hat{q}}{\cos\quad\alpha_{2}}};} - {\Delta\quad{\overset{.}{\Omega}}_{E\quad 0}} - {r\quad\Delta\quad\Omega_{N\quad 0}} + {\omega_{0}^{2}\Delta\quad\hat{\alpha}}} = {\frac{\Delta\quad\overset{\sim}{q}\omega_{0}}{n\quad\cos\quad\alpha_{2}} + \frac{\Delta\quad{\hat{W}}_{N}}{R\quad\cos\quad\alpha_{2}}}};} - {\Delta\quad\overset{\hat{.}}{\alpha}} + {r\quad{\hat{\beta}}_{1}} - {n^{2}\cos^{2}\alpha_{2}{\Delta\Omega}_{E\quad 0}}} = {{\Delta\quad\hat{p}} + \frac{n\quad\Delta\quad{\overset{\sim}{W}}_{E}}{\omega_{0}R} + {\left( {1 - n^{2}} \right)\cos^{2}\alpha\quad\frac{V}{R}\Delta\quad K}}} & (2)         \end{matrix}$

For the strapdown variant Δ{tilde over (p)}=0; Δ{tilde over (q)}=0; Δ{tilde over (W)}_(E)=0.

As it follows from these equations, the natural frequencies of the system ω₀ cos α and nω₀ cos α differ from Schuler frequency ω₀.

nω₀ cos α—directing force to the meridian.

The plane of ring 18 can be maintained perpendicularly to the compass meridian, controlling motor 20 in the following modes:

1) {dot over (K)}=F(β₂−β₁)

where

-   -   {dot over (K)}—speed of motor 20 control action in response to a         signal,     -   F—transfer function.

2) controlling in response to a signal from compass heading generated by the inertial system with integral correction;

3) controlling in response to a compass heading signal from the external heading indicator.

By measurement (β₂−β₁) or by ΔΩ_(E0) we can estimate the eastern drift of the gyroscopic unit of the strapdown inertial system or determine an error of compass heading reproduced by motor 20 accurate to ${\Delta\quad\Delta\quad K} = {- {\frac{\Delta\quad\hat{p}}{\frac{V}{R}\left( {n\quad\cos\quad\alpha} \right)^{2}}.}}$

The observed generalized coordinate (β₂−β₁) or ΔΩ_(E0) make it possible to introduce autonomously the dissipative forces into system control signals and thus provide autonomous damping—asymptotic stability of the system.

It can be shown that using the estimates of angles β₂ and β₁, as well as the accelerometer readings W_(E1); W_(E2) and W_(E) _(IS) it is possible to measure the observable generalized coordinates {circumflex over (β)} and β_(IS), where β_(IS)—an error in stabilization of gyro horizon of the inertial system with integral correction, W_(E) _(IS) —a reading of the corresponding accelerometer of the inertial system with integral correction.

FIG. 4 presents the block diagram of an inertial system with linear correction with gyro platforms in two-axis gimbal suspension.

Let the right coordinate systems E₁N₁ζ₁ and E₂N₂ζ₂ be bound with the gyro platforms. Coordinate systems E₁N₁ζ₁ and E₂N₂ζ₂ will be obtained from coordinate system E₀N₀ζ₀ by turning sequentially:

1) around axis Oζ₀ at angles ΔK₁ and ΔK₂, respectively;

2) around axes OE₁ and OE₂ at angles α₁ and α₂, respectively.

Projections of absolute angular rate of trihedrals E₁N₁ζ₁ and E₂N₂ζ₂ on their axes will be the following, with allowance for gyro drifts: $\Omega_{E\quad 1} = {{\overset{.}{\alpha}}_{1} + {\frac{V}{R}\Delta\quad K_{1}} + {\Delta\quad p_{1}}}$ $\Omega_{N\quad 1} = {{\Delta\quad{\overset{.}{K}}_{1}\sin\quad\alpha_{1}} + {\frac{V}{R}\cos\quad\alpha_{1}} + {r\quad\sin\quad\alpha_{1}} + {\beta_{IS}\cos\quad\alpha_{1}} + {\Delta\quad q_{1}}}$ $\Omega_{E\quad 2} = {{\overset{.}{\alpha}}_{2} + {\frac{V}{R}\Delta\quad K_{2}} + {\Delta\quad p_{2}}}$ $\Omega_{N\quad 2} = {{\Delta\quad{\overset{.}{K}}_{2}\sin\quad\alpha_{2}} + {\frac{V}{R}\cos\quad\alpha_{2}} + {r\quad\sin\quad\alpha_{2}} + {{\overset{.}{\beta}}_{IS}\cos\quad\alpha_{2}} + {\Delta\quad q_{2}}}$

Accelerometer signals along axes OE₁; ON₁ and axes OE₂; ON₂ will be the following: W _(E1) =−V−rVΔK ₁ +gβ _(IS) +ΔW _(E1) W _(N1)=−(rV cos α₁ +g sin α₁)+ΔW _(N1) W _(E2) =−V−rVΔK ₂ +gβ _(IS) +ΔW _(E2) W _(N2)=−(rV cos α₂ +g sin α₂)+ΔW _(N2)

In order to fulfill the conditions ${{tg}\quad\alpha_{1}} = {{{- \frac{n}{\omega_{0}}}\frac{V}{R}\quad{and}\quad{tg}\quad\alpha_{2}} = {\frac{n}{\omega_{0}}\frac{V}{R}}}$

we'll define the following expressions for control signals of the gyroscopes of the first and the second gyro pendulums: $\Omega_{E\quad 1\quad{{con}.}} = {\frac{n}{\omega_{0}}\left( {\frac{W_{E\quad 1}}{R} + {\frac{{\overset{.}{V}}_{{dev}{.1}}}{R}\sin^{2}\alpha_{{dev}{.1}}}} \right)}$ $\Omega_{N\quad 1\quad{{con}.}} = {{{- \frac{\omega_{0}}{n}}\sin\quad\alpha_{{dev}{.1}}} + {\frac{n}{\omega_{0}}\frac{\left( {W_{N\quad 1} + {g\quad\sin\quad\alpha_{{dev}{.1}}}} \right)}{R}}}$ $\Omega_{E\quad 2\quad{{con}.}} = {\frac{n}{\omega_{0}}\left( {\frac{W_{E\quad 2}}{R} + {\frac{{\overset{.}{V}}_{{dev}{.2}}}{R}\sin^{2}\alpha_{{dev}{.2}}}} \right)}$ $\Omega_{N\quad 2\quad{{con}.}} = {{\frac{\omega_{0}}{n}\sin\quad\alpha_{{dev}{.2}}} - {\frac{n}{\omega_{0}}\frac{\left( {W_{N\quad 2} + {g\quad\sin\quad\alpha_{{dev}{.2}}}} \right)}{R}}}$ where ${{\frac{n}{\omega_{0}}\frac{{\overset{.}{V}}_{{dev}{.1}}}{R}} = \frac{{\overset{.}{\alpha}}_{{dev}{.1}}}{\cos^{2}\alpha_{1}}};{{\frac{n}{\omega_{0}}\frac{{\overset{.}{V}}_{{dev}{.2}}}{R}} = \frac{{\overset{.}{\alpha}}_{{dev}{.2}}}{\cos^{2}\alpha_{2}}};$ α_(dev.1) = α₁ − Δ  α_(IS); α_(dev.2) = α₂ − Δ  α_(IS).

Then the equations of motion (operation) of two gyro platforms will be the following: Ω_(E1)=Ω_(E1con.) Ω_(E2)=Ω_(E2con.) Ω_(N1)=Ω_(N1con.) Ω_(N2)=Ω_(N2con.)

The error equations for the strapdown inertial system with linear correction will have the form: ${{{- \Delta}\quad{\overset{\overset{.}{\hat{}}}{\Omega}}_{N}} + {r\quad\Delta\quad{\hat{\Omega}}_{E}} - {\frac{\omega_{0}}{n}\Delta\quad{\overset{\sim}{\Omega}}_{E}}} = {{{\omega_{0}^{2}\frac{\Delta\quad W_{E}}{g}} + {\omega_{0}^{2}\beta_{IS}} - {\Delta\quad{\overset{\overset{.}{\sim}}{\Omega}}_{N}} + {r\quad\Delta\quad{\hat{\Omega}}_{E}} - {\frac{\omega_{0}}{n}\Delta\quad{\hat{\Omega}}_{E}}} = {{{\frac{\omega_{0}}{n}\Delta\quad p} + {\frac{\omega_{0}}{n}{tg}^{2}{\alpha\Delta}\quad{\overset{.}{\alpha}}_{IS}} - {\Delta\quad{\overset{\overset{.}{\sim}}{\Omega}}_{E}} - {r\quad\Delta\quad{\overset{\sim}{\Omega}}_{N}} + {\frac{\omega_{0}}{n}\Delta\quad{\hat{\Omega}}_{N}}} = {{{\frac{\omega_{0}}{n}\frac{\Delta\quad q}{\cos\quad\alpha}} + {\frac{\omega_{0}}{n}{\overset{.}{\beta}}_{IS}} - {\Delta\quad{\overset{\overset{.}{\hat{}}}{\Omega}}_{E}} - {r\quad\Delta\quad{\hat{\Omega}}_{N}} + {\frac{\omega_{0}}{n}\Delta\quad{\overset{\sim}{\Omega}}_{N}}} = {\frac{\omega_{0}^{2}\Delta\quad W_{N}}{g\quad\cos\quad\alpha} - {{\omega_{0}^{2}\left( {1 - \frac{1}{n^{2}}} \right)}\Delta\quad\alpha_{IS}}}}}}$ where $\begin{matrix} {{{\Delta\quad{\hat{\Omega}}_{N}} = {\frac{\omega_{0}}{2\quad n}\frac{\left( {{\Delta\quad\alpha_{2}} - {\Delta\quad\alpha_{1}}} \right)}{\cos^{2}\alpha}}};} & {{{\Delta\quad{\hat{\Omega}}_{E}} = {\frac{V}{R}{\Delta\left( \frac{K_{2} + K_{1}}{2} \right)}}};} \\ {{{\Delta\quad{\hat{\Omega}}_{N}} = {\frac{\omega_{0}}{2\quad n}\frac{\left( {{\Delta\quad\alpha_{2}} + {\Delta\quad\alpha_{1}}} \right)}{\cos^{2}\alpha}}};} & {{{\Delta\quad{\hat{\Omega}}_{E}} = {\frac{V}{R}\frac{\left( {K_{2} - K_{1}} \right)}{2}}};} \end{matrix}$ Δ{tilde over (Ω)}_(E)—observed generalized coordinate.

With the relevant control signals the equations for free oscillations for inertial system in two-axes gimbal will be the following: ${{{- \Delta}\quad{\overset{\overset{.}{\hat{}}}{\Omega}}_{N}} + {r\quad\Delta\quad{\hat{\Omega}}_{E}} - {\left( {P_{1} + \frac{\omega_{0}}{n}} \right)\Delta\quad{\overset{\sim}{\Omega}}_{E}}} = {{0 - {\Delta\quad{\overset{\overset{.}{\sim}}{\Omega}}_{N}} + {\left( {P_{2} + r} \right)\Delta\quad{\overset{\sim}{\Omega}}_{E}} - {\frac{\omega_{0}}{n}\Delta\quad{\hat{\Omega}}_{E}}} = {{0 - {\Delta\quad{\overset{\overset{.}{\sim}}{\Omega}}_{E}} + {P_{3}\quad\Delta\quad{\overset{\sim}{\Omega}}_{E}} - {r\quad\Delta\quad{\overset{\sim}{\Omega}}_{N}} + {\frac{\omega_{0}}{n}\Delta\quad{\hat{\Omega}}_{N}}} = {{0 - {\Delta\quad{\overset{\overset{.}{\sim}}{\Omega}}_{E}} - {r\quad\Delta\quad{\hat{\Omega}}_{N}} + {\frac{\omega_{0}}{n}\Delta\quad{\overset{\sim}{\Omega}}_{N}} + {P_{4}\Delta\quad{\overset{\sim}{\Omega}}_{E}}} = 0}}}$ where

P1; P2; P3; P4—transfer functions.

If we take P1; P2; P3; P4 as constant values, we can obtain a characteristic equation in the following form: Δ=S ⁴ +b ₃ S ³ +b ₂ S ² +b ₁ S+b ₀, where b₀; b₁; b₂; b₃—the constants independent of each other.

Thus autonomous damping can be provided—asymptotic stability of Schuler loop of the inertial system.

With the strapdown version of joint operation of the inertial system with linear correction and the inertial system with integral correction we obtain a multiparameter autonomous set of inertial systems, because simultaneous operation of several inertial systems can be virtually provided in different modes with different dynamic properties, which also ensure asymptotic stability of the system with different parameters n, hence with different values of speed deviation α. This condition allows instrumental errors of SINS sensors to be estimated continuously during operation, comparing the similar generated output values of different systems between each other.

As an example, let us consider the mechanism of estimating he most vulnerable component of gyro system drift—drift around the vertical axis—Δr (see equations (3)). Let's assume ${{\Delta\quad\Omega_{N\quad 0}} = \frac{\Delta\quad\hat{q}}{\cos\quad\alpha_{2}}},{{{then}\quad\Delta\quad r_{{dev}.}} = \frac{\Omega_{N\quad 0_{2}} - \Omega_{N\quad 0_{1}}}{{{tg}\quad\alpha_{2}} - {{tg}\quad\alpha_{1}}}},$ where

Δr_(dev.)—an estimate of strapdown gyro system drift around the vertical axis;

Ω_(N0) ₂ ; Ω_(N0) ₁ —values of horizontal components of absolute angular rate of the vehicle, generated by two inertial systems with linear correction with different values of parameter n. 

1. The method of generation of navigation parameters and position vertical includes: measurement of specific force components using accelerometers, sensitivity axes of which are connected with the gyro platform; generation of gyro platform control signals; following to indicate the generated signals using a gyro or absolute angular rate sensors, or measurement of specific force components using accelerometers; measurement of signals from gyros or absolute angular rate sensors, sensitivity axes of which are directed along axes of the instrumental trihedral; analytical solution of the orientation problem by means of inertial system operation simulation; generation of navigation parameters and position vertical, and differs by the fact that control signals of a gyro platform or a model of the gyro platform of the inertial system with linear correction are generated under the condition that in the absence of ballistic deviations a natural period of the gyro platform or the gyro platform model differs from Schuler period.
 2. The method of generation of navigation parameters and position vertical includes: measurement of specific force components using accelerometers, sensitivity axes of which are connected with the gyro platform; generation of gyro platform control signals; following to indicate the generated signals using a gyro or absolute angular rate sensors, or measurement of specific force components using accelerometers; measurement of signals from gyros or absolute angular rate sensors, sensitivity axes of which are directed along axes of the instrumental trihedral; analytical solution of the orientation problem by means of inertial system operation simulation; generation of navigation parameters and position vertical, and differs by the fact that control signals of gyro platforms or models of the gyro platforms of the inertial system with linear correction are generated to provide, in the absence of ballistic deviations, either nonlinear relation between a value of speed deviation and a value of horizontal component of the vehicle absolute angular rate, or the relation of type $\alpha = {\frac{n}{\omega_{0}}\frac{V}{R}}$  at small values of speed deviations, with which relations tgα=sin α=α are admissible.
 3. The method of generation of navigation parameters and position vertical includes: measurement of specific force components using accelerometers, sensitivity axes of which are connected with the gyro platform; generation of gyro platform control signals; following to indicate the generated signals using a gyro or absolute angular rate sensors, or measurement of specific force components using accelerometers; measurement of signals from gyros or absolute angular rate sensors, sensitivity axes of which are directed along axes of the instrumental trihedral; analytical solution of the orientation problem by means of inertial system operation simulation; generation of navigation parameters and position vertical, and differs by the fact that asymptotic stability (autonomous damping) of each inertial system or each inertial system model, including (if necessary) the inertial system with integral correction, is provided using differences of the similar data generated by the inertial systems with linear correction, as well as the inertial system with integral correction (if necessary) or their models with different natural periods, and (or) using values of observable generalized coordinates of the system or its model.
 4. The method of generation of navigation parameters and position vertical includes: measurement of specific force components using accelerometers, sensitivity axes of which are connected with the gyro platform; generation of gyro platform control signals; following to indicate the generated signals using a gyro or absolute angular rate sensors, or measurement of specific force components using accelerometers; measurement of signals from gyros or absolute angular rate sensors, sensitivity axes of which are directed along axes of the instrumental trihedral; analytical solution of the orientation problem by means of inertial system operation simulation; generation of navigation parameters and position vertical, and differs by the fact that estimation of instrumental errors is provided using differences of the similar data generated by inertial systems with linear correction, as well as inertial systems with integral correction (if necessary), or their models with different natural periods, with different parameters n and in different modes.
 5. The method of generation of navigation parameters and position vertical includes: measurement of specific force components using accelerometers, sensitivity axes of which are connected with the gyro platform; generation of gyro platform control signals; following to indicate the generated signals using a gyro or absolute angular rate sensors, or measurement of specific force components using accelerometers; measurement of signals from gyros or absolute angular rate sensors, sensitivity axes of which are directed along axes of the instrumental trihedral; analytical solution of the orientation problem by means of inertial system operation simulation; generation of navigation parameters and position vertical, and differs by the fact that accuracy of the vehicle heading generation is increased and the system readiness time is reduced due to increasing directing force to the compass meridian in the inertial system with linear correction. 